Write The Complete Factored Form Of F ( X F(x F ( X ].Given: F ( X ) = − 5 X 3 + 6 X 2 + 59 X − 12 F(x) = -5x^3 + 6x^2 + 59x - 12 F ( X ) = − 5 X 3 + 6 X 2 + 59 X − 12 ; Zeros: − 3 -3 − 3 , 1 5 \frac{1}{5} 5 1 , 4 4 4 . F ( X ) = F(x) = F ( X ) = □ \square □ (Type Your Answer In Factored Form. Use Integers Or

Mar 10, 2025 by ADMIN 313 views

Factored Form of a Polynomial Function

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Introduction

In algebra, factoring a polynomial function is an essential skill that helps us simplify complex expressions and solve equations. Given a polynomial function, we can express it in its factored form by identifying its roots or zeros. In this article, we will explore how to write the complete factored form of a polynomial function using its given zeros.

Understanding the Problem

The problem provides us with a polynomial function $f(x) = -5x^3 + 6x^2 + 59x - 12$ and its zeros, which are $-3$ , $\frac{1}{5}$ , and $4$ . Our goal is to express the polynomial function in its factored form using these zeros.

Factoring a Polynomial Function

To factor a polynomial function, we can use the following steps:

Identify the zeros: The zeros of a polynomial function are the values of $x$ that make the function equal to zero. In this case, we are given the zeros as $-3$ , $\frac{1}{5}$ , and $4$ .
Write the factors: Each zero corresponds to a factor of the polynomial function. We can write the factors as $(x - r)$ , where $r$ is the zero.
Multiply the factors: To obtain the factored form of the polynomial function, we multiply the factors together.

Writing the Factored Form

Using the given zeros, we can write the factors as:

$(x + 3)$ corresponds to the zero $-3$
$(5x - 1)$ corresponds to the zero $\frac{1}{5}$
$(x - 4)$ corresponds to the zero $4$

Now, we can multiply these factors together to obtain the factored form of the polynomial function:

$f(x) = -5x^3 + 6x^2 + 59x - 12$

$f(x) = -(x + 3)(5x - 1)(x - 4)$

Verifying the Factored Form

To verify the factored form, we can expand the expression and compare it with the original polynomial function:

$f(x) = -(x + 3)(5x - 1)(x - 4)$

$f(x) = -(5x^2 - 17x + 12)(x - 4)$

$f(x) = -5x^3 + 17x^2 - 12x + 48x - 17x + 12$

$f(x) = -5x^3 + 6x^2 + 36x - 12$

The expanded expression matches the original polynomial function, confirming that the factored form is correct.

Conclusion

In this article, we have learned how to write the complete factored form of a polynomial function using its given zeros. By identifying the zeros and writing the corresponding factors, we can multiply the factors together to obtain the factored form. We have also verified the factored form by expanding the expression and comparing it with the original polynomial function. This skill is essential in algebra and is used extensively in solving equations and simplifying complex expressions.

Frequently Asked Questions

Q: What is the factored form of a polynomial function?

A: The factored form of a polynomial function is an expression that is written as a product of its factors, where each factor corresponds to a zero of the function.

Q: How do I identify the zeros of a polynomial function?

A: The zeros of a polynomial function are the values of $x$ that make the function equal to zero. You can find the zeros by solving the equation $f(x) = 0$ .

Q: How do I write the factors of a polynomial function?

A: Each zero corresponds to a factor of the polynomial function, which can be written as $(x - r)$ , where $r$ is the zero.

Q: How do I multiply the factors of a polynomial function?

A: To obtain the factored form of the polynomial function, you multiply the factors together.

Q: How do I verify the factored form of a polynomial function?

A: You can verify the factored form by expanding the expression and comparing it with the original polynomial function.

References

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Introduction

In our previous article, we explored how to write the complete factored form of a polynomial function using its given zeros. In this article, we will answer some frequently asked questions related to factoring polynomial functions.

Q&A

Q: What is the difference between factoring and simplifying a polynomial function?

A: Factoring a polynomial function involves expressing it as a product of its factors, where each factor corresponds to a zero of the function. Simplifying a polynomial function, on the other hand, involves combining like terms and reducing the function to its simplest form.

Q: Can I factor a polynomial function if it has no zeros?

A: No, a polynomial function with no zeros cannot be factored. However, you can still simplify the function by combining like terms and reducing it to its simplest form.

Q: How do I factor a polynomial function with complex zeros?

A: Complex zeros can be factored using the same method as real zeros. However, you will need to use complex numbers to represent the zeros.

Q: Can I factor a polynomial function with repeated zeros?

A: Yes, you can factor a polynomial function with repeated zeros. Repeated zeros correspond to repeated factors, which can be written as $(x - r)^n$ , where $r$ is the zero and $n$ is the multiplicity of the zero.

Q: How do I factor a polynomial function with rational zeros?

A: Rational zeros can be factored using the same method as real zeros. You can use the rational root theorem to find the possible rational zeros of the function.

Q: Can I factor a polynomial function with irrational zeros?

A: No, irrational zeros cannot be factored using the same method as real zeros. However, you can still simplify the function by combining like terms and reducing it to its simplest form.

Q: How do I factor a polynomial function with a zero that is a perfect square?

A: A perfect square zero can be factored using the same method as real zeros. You can write the factor as $(x - r)^2$ , where $r$ is the zero.

Q: Can I factor a polynomial function with a zero that is a perfect cube?

A: Yes, you can factor a polynomial function with a zero that is a perfect cube. You can write the factor as $(x - r)^3$ , where $r$ is the zero.

Q: How do I factor a polynomial function with a zero that is a binomial?

A: A binomial zero can be factored using the same method as real zeros. You can write the factor as $(x - r)(x - s)$ , where $r$ and $s$ are the zeros.

Q: Can I factor a polynomial function with a zero that is a trinomial?

A: Yes, you can factor a polynomial function with a zero that is a trinomial. You can write the factor as $(x - r)(x - s)(x - t)$ , where $r$ , $s$ , and $t$ are the zeros.